I've heard they used everything from tape to velcro (Antron fleece can act as velcro).
Also, if it's a large/heavy enough prop (such as the earpiece for the telephone in TMM) they'll attach a rod to the object and hold it up along with the arm rod.
One idea for tackling this problem that I got from the posable dolls they sell is to line the inside of the hand with a wire frame. That way you can bend the fingers around anything light enough for the puppet to hold. I believe the Muppets incorporated this method too (look at Kermit's hand in TGMC when he is reaching to catch the Baseball Diamond — while he's not holding anything, his fingers are cupped and stiff.)